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{{Infobox MOS
{{Interwiki
| Name = smitonic
|en=4L 3s
| Periods = 1
|es=
| nLargeSteps = 4
|de=
| nSmallSteps = 3
|ja=4L 3s
| Equalized = 2
| Paucitonic = 1
| Pattern = LLsLsLs
}}
}}
{{Infobox MOS}}


'''4L 3s''' refers to an [[MOS]] scale with four large steps and three small steps, one mode of which is '''LLsLsLs'''. It is generated by any interval between 1\4edo (one degree of [[4edo]], or 300¢) and 2\7edo (two degrees of [[7edo]], or approx. 342.857¢).
{{MOS intro}}
4L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], where one large step of diatonic ([[5L 2s]]) is replaced with a small step.


4L 3s can be thought of as a [[Warped diatonic|warped diatonic scale]], because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).
== Name ==
{{TAMNAMS name}}


== Standing assumptions ==
== Scale properties ==
The [[TAMNAMS]] system is used in this article to name 4L 3s intervals and step size ratios and step ratio ranges.
{{TAMNAMS use}}


The notation used in this article is LsLsLsL = JKLMNOPJ unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)
=== Intervals ===
{{MOS intervals}}


Thus the [[11edo]] gamut is as follows:
=== Generator chain ===
{{MOS genchain}}


'''J''' J&/K@ '''K''' '''L''' L&/M@ '''M''' '''N''' N&/O@ '''O''' '''P''' P&/J@ '''J'''
=== Modes ===
{{MOS mode degrees}}


== Names ==
==== Proposed names ====
The [[TAMNAMS]] MOS naming system (used in this article) uses the name '''smitonic''' ''smy-TON-ik'' /smaɪˈtɒnɪk/ for this pattern. The name is derived from 'sharp minor third', since the central range of the spectrum, 4\15 = 320¢ to 7\18 = 333.33¢, has minor third generators that are significantly sharp of 6/5.
Alexandru Ianu ([[User:Ayceman|Ayceman]])<ref>Description of ''Sylvian Moon Dance'' mentioning the naming proposal https://musescore.com/user/36772625/scores/6700443 – The theme relates to the mystical nature of the Tribunal and TES lore, which fits smitonic.</ref> has proposed the following mode names relating to the Almsivi in Morrowind (TES):
{{MOS modes
| Mode Names=Nerevarine $
Vivecan $
Lorkhanic $
Sothic $
Kagrenacan $
Almalexian $
Dagothic $
}}


== Intervals ==
== Theory ==
Note: In TAMNAMS, a k-step interval class in smitonic may be called a "k-step", "k-mosstep", or "k-smistep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
=== Low harmonic entropy scales ===
There are two notable harmonic entropy minima:
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1.
* [[myna|Myna temperament]], in which the generator is also 6/5 but it takes 10 of them to make a 6/1, meaning that a larger MOS than 4L&nbsp;3s is required to reach 3/2 or 4/3.


{| class="wikitable center-all"
=== Temperament interpretations ===
|-
{{main|4L&nbsp;3s/Temperaments}}
! Generators
4L&nbsp;3s has the following temperament interpretations:
! Notation (1/1 = J)
* [[Sixix]], with generators around 338.6{{c}}.
! [[TAMNAMS]] name
* [[Orgone]], with generators around 323.4{{c}}.
! In L's and s's
* [[Kleismic]], with generators around 317{{c}}.
! Generators
! Notation of 2/1 inverse
! [[TAMNAMS]] name
! In L's and s's
|-
| colspan="8" style="text-align:left" | The 7-note MOS has the following intervals (from some root):
|-
| 0
| J
| perfect unison
| 0L + 0s
| 0
| J
| octave
| 4L + 3s
|-
| 1
| L
| perfect 2-mosstep
| 1L + 1s
| -1
| O
| perfect 5-mosstep
| 3L + 2s
|-
| 2
| N
| minor 4-mosstep
| 2L + 2s
| -2
| M
| major 3-mosstep
| 2L + 1s
|-
| 3
| P
| minor 6-mosstep
| 3L + 3s
| -3
| K
| major 1-mosstep
| 1L + 0s
|-
| 4
| K@
| minor 1-mosstep
| 0L + 1s
| -4
| Q&
| major 6-mosstep
| 4L + 2s
|-
| 5
| M@
| minor 3-mosstep
| 1L + 2s
| -5
| N&
| major 4-mosstep
| 3L + 1s
|-
| 6
| O@
| diminished 5-mosstep
| 2L + 3s
| -6
| L&
| augmented 2-mosstep
| 2L + 0s
|-
| colspan="8" style="text-align:left" | The chromatic 11-note MOS (either [[7L 4s]], [[4L 7s]], or [[11edo]]) also has the following intervals (from some root):
|-
| 7
| J@
| diminished 7-mosstep
| 5L + 2s
| -7
| J&
| augmented mosunison; chroma
| 1L - 1s
|-
| 8
| L@
| diminished 2-mosstep
| 0L + 2s
| -8
| O&
| augmented 5-mosstep
| 4L + 1s
|-
| 9
| N@
| diminished 4-mosstep
| 1L + 3s
| -9
| M&
| augmented 3-mosstep
| 3L + 0s
|-
| 10
| P@
| diminished 6-mosstep
| 2L + 4s
| -10
| K&
| augmented 1-mosstep
| 2L - 1s
|}


== Low harmonic entropy scales ==
Other temperaments, such as [[amity]] and [[myna]], require more than 7 pitches to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.
There are two notable harmonic entropy minima:
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1 (making the diminished 5-mosstep 3/2)
* [[Myna]], in which the generator is also 6/5 but now '''10''' of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).


== Tuning ranges ==
== Tuning ranges ==
{{Todo|Populate|comment=Populate with JI ratios from prior edits of this page.|inline=1}}
=== Simple tunings ===
=== Simple tunings ===
{| class="wikitable right-2 right-3 right-4 sortable "
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 11edo, 15edo, and 18edo, respectively.
|-
{{MOS tunings}}
! class="unsortable"|Degree
! [[11edo]] (basic)
! [[15edo]] (hard)
! [[18edo]] (soft)
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-bgcolor="#eaeaff"
| unison
| 0\11, 0.0
| 0\15, 0.0
| 0\18, 0.0
| J
|
| 0
|-
| minor 1-mosstep
| 1\11, 109.1
| 1\15, 80.0
| 2\18, 133.3
| K@
|
| +4
|-
| major 1-mosstep
| 2\11, 218.2
| 3\15, 240.0
| 3\18, 200.0
| K
| 8/7
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 3\11, 327.3
| 4\15, 320.0
| 5\18, 333.3
| L
| 77/64, 6/5
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 4\11, 436.4
| 6\15, 480.0
| 6\18, 400.0
| L&
|
| -6
|-
| minor 3-mosstep
| 4\11, 436.4
| 5\15, 400.0
| 7\18, 466.7
| M@
| 14/11
| +5
|-
| major 3-mosstep
| 5\11, 545.5
| 7\15, 560.0
| 8\18, 533.3
| M
| 11/8
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 6\11, 656.6
| 8\15, 640.0
| 10\18, 666.7
| N
| 16/11
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 7\11, 763.6
| 10\15, 800.0
| 11\18, 733.3
| N&
| 11/7
| -5
|-
| dim. 5-mosstep
| 7\11, 763.6
| 9\15, 720.0
| 12\18, 800.0
| O@
|
| +6
|-
| perf. 5-mosstep
| 8\11, 872.7
| 11\15, 880.0
| 13\18, 866.7
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 9\11, 981.8
| 12\15, 960.0
| 15\18, 1000.0
| P
| 7/4
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 10\11, 1090.9
| 14\15, 1120.0
| 16\18, 1066.7
| P&
|
| -4
|}


=== Parasoft ===
=== Parasoft tunings ===
[[Parasoft]] smitonic tunings have step ratios between 4/3 and 3/2, which implies a generator sharper than 5\18 = 333.3¢ and flatter than 7\25 = 336.0¢.
Parasoft smitonic tunings can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:


Parasoft smitonic can be considered "meantone smitonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:
* The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
* The large step is a "meantone", somewhere between near-10/9 (as in [[32edo]]) and near-9/8 (as in [[18edo]]).
* The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.
* The augmented 2-mosstep (made of two large steps) is a roughly [[meantone]]-sized major third, thus is a stand-in for the classical diatonic major third.
Parasoft smitonic tunings have both minor fifths and major fifths about equally off a just fifth, and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.


Parasoft smitonic EDOs include [[18edo]], [[25edo]], and [[43edo]].
These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702{{c}}), and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.
* 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted [[19edo]] diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect 5-mosstep) and falling fifths (666.7c, a major 4-mosstep) almost equally off from a just fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* [[25edo]] can be used to make the augmented 2-mosstep a good [[5/4]] (384¢).


The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.
Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[18edo]] (soft)
! [[25edo]] (supersoft)
! [[43edo]]
! Optimized (2.9.5 [[POTE]] [[Dual-fifth temperaments|dual-3 sixix]]) tuning
|-
| generator (g)
| 5\18, 333.3
| 7\25, 336.0
| 12\43, 334.9
| 335.84
|-
| L (octave - 3g)
| 3\18, 200.0
| 4\25, 192.0
| 7\43, 195.3
| 193.16
|-
| s (4g - octave)
| 2\18, 133.3
| 3\25, 144.0
| 5\43, 139.5
| 143.36
|}


==== Intervals ====
* 18edo can be used to make the large and small steps more distinct, or can be considered a distorted 19edo diatonic.
Sortable table of the extended generator chain (-13 to 13 generators) in parasoft smitonic tunings. The several different interval flavors separated by the chroma shows that parasoft smitonic is a useful [[cluster MOS]], though many of the intervals lack simple JI interpretations.
** 18edo has a major 1-mosstep that is close to 9/8 (203{{c}}).
{| class="wikitable right-2 right-3 right-4 sortable "
** 18edo's major and minor 4-mossteps are both equally off from 12edo's diatonic perfect 5th (700{{c}}) by 33.3{{c}}.
|-
** 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
! class="unsortable"|Degree
* The augmented 2-mosstep of 25edo is very close to 5/4 (386{{c}}).
! [[18edo]] (soft)
{{MOS tunings|Step Ratios=3/2; 7/5; 4/3}}
! [[25edo]] (supersoft)
! [[43edo]]
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-bgcolor="#eaeaff"
| | unison
| 0\18, 0.0
| 0\25, 0.0
| 0\43, 0.0
| J
| 1/1
| 0
|-bgcolor="#eaeaff"
| chroma
| 1\18, 66.7
| 1\25, 48.0
| 2\43, 55.8
| J&
|
| -7
|-
| dim. 1-mosstep
| 1\18, 66.7
| 2\25, 96.0
| 3\43, 83.7
| K@@
|
| +11
|-
| minor 1-mosstep
| 2\18, 133.3
| 3\25, 144.0
| 5\43, 139.5
| K@
| 13/12
| +4
|-
| major 1-mosstep
| 3\18, 200.0
| 4\25, 192.0
| 7\43, 195.3
| K
| 9/8, 10/9
| -3
|-
| aug. 1-mosstep
| 4\18, 266.7
| 5\25, 240.0
| 9\43, 251.2
| K&
|
| -10
|-bgcolor="#eaeaff"
| dim. 2-mosstep
| 4\18, 266.7
| 6\25, 288.0
| 10\43, 279.1
| L@
|
| +8
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 5\18, 333.3
| 7\25, 336.0
| 12\43, 334.9
| L
| 17/14, 40/33
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 6\18, 400.0
| 8\25, 384.4
| 14\43, 390.7
| L&
| 5/4
| -6
|-bgcolor="#eaeaff"
| doubly aug. 2-mosstep
| 7\18, 466.7
| 9\25, 432.0
| 16\43, 446.5
| L&&
|
| -13
|-
| dim. 3-mosstep
| 6\18, 400.0
| 9\25, 432.0
| 15\43, 418.6
| M@@
|
| +12
|-
| minor 3-mosstep
| 7\18, 466.7
| 10\25, 480.0
| 17\43, 474.4
| M@
| 21/16
| +5
|-
| major 3-mosstep
| 8\18, 533.3
| 11\25, 528.0
| 19\43, 530.2
| M
| 19/14, 34/25
| -2
|-
| aug. 3-mosstep
| 9\18, 600.0
| 12\25, 576.0
| 21\43, 586.0
| M&
| 7/5
| -9
|-bgcolor="#eaeaff"
| dim. 4-mosstep
| 9\18, 600.0
| 13\25, 624.0
| 22\43, 614.0
| N@
| 10/7
| +9
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 10\18, 666.7
| 14\25, 672.0
| 24\43, 669.8
| N
| 28/19, 25/17
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 11\18, 733.3
| 15\25, 720.0
| 26\43, 725.6
| N&
| 32/21
| -5
|-bgcolor="#eaeaff"
| aug. 4-mosstep
| 12\18, 800.0
| 16\25, 768.0
| 28\43, 781.4
| N&&
|
| -12
|-
| doubly dim. 5-mosstep
| 11\18, 733.3
| 16\25, 768.0
| 27\43, 753.5
| O@@
|
| +13
|-
| dim. 5-mosstep
| 12\18, 800.0
| 17\25, 816.0
| 29\43, 809.3
| O@
| 8/5
| +6
|-
| perf. 5-mosstep
| 13\18, 866.7
| 18\25, 864.0
| 31\43, 865.1
| O
| 28/17, 33/20
| -1
|-
| aug. 5-mosstep
| 14\18, 933.3
| 19\25, 912.0
| 33\43, 920.9
| O&
|
| -8
|-bgcolor="#eaeaff"
| dim. 6-mosstep
| 14\18, 933.3
| 20\25, 960.0
| 34\34, 948.8
| P@
|
| +10
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 15\18, 1000.0
| 21\25, 1008.0
| 36\43, 1004.7
| P
| 16/9, 9/5
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 16\18, 1066.7
| 22\25, 1056.0
| 38\43, 1060.5
| P&
| 24/13
| -4
|-bgcolor="#eaeaff"
| aug. 6-mosstep
| 17\18, 1133.3
| 23\25, 1104.0
| 40\43, 1116.3
| P&
|
| -11
|-
| dim. mosoctave
| 17\18, 1133.3
| 24\25, 1152.0
| 41\43, 1144.2
| J@
|
| +7
|}


=== Hyposoft ===
=== Hyposoft tunings ===
[[Hyposoft]] tunings of smitonic  have [[step ratio]]s between 3/2 and 2/1 which implies that the generator is a supraminor third sharper than 3\11 = 327.27¢ and flatter than 5\18 = 333.33¢.
Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327{{c}} and 333{{c}}. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic".


The large step is a sharper major second in these tunings than in parasoft tunings. These tunings could be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic", in analogy to parasoft smitonic being meantone smitonic.
Edos include [[11edo]] (not shown), [[18edo]], and [[29edo]].


{| class="wikitable right-2 right-3 right-4 right-5"
{{MOS tunings|Step Ratios=3/2; 5/3; 7/4}}
|-
!
! [[11edo]] (basic)
! [[18edo]] (soft)
! [[29edo]] (semisoft)
|-
| generator (g)
| 3\11, 327.27
| 5\18, 333.33
| 8\29, 331.03
|-
| L (octave - 3g)
| 2\11, 218.18
| 3\18, 200.00
| 5\29, 206.90
|-
| s (4g - octave)
| 1\11, 109.09
| 2\18, 133.33
| 3\29, 124.14
|}
==== Intervals ====
Sortable table of major and minor intervals in hyposoft smitonic tunings (11edo and 18edo not shown):
{| class="wikitable right-2 sortable "
|-
! class="unsortable"|Degree
! [[29edo]] (semisoft)
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios (for 29edo)
! #Gens up
|-bgcolor="#eaeaff"
| unison
| 0\29, 0.0
| J
| 1/1
| 0
|-
| minor 1-mosstep
| 3\29, 124.1
| K@
| 14/13
| +4
|-
| major 1-mosstep
| 5\29, 206.9
| K
| 9/8
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 8\29, 331.0
| L
| 23/19, 40/33
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 10\29, 413.8
| L&
| 14/11
| -6
|-
| minor 3-mosstep
| 11\29, 455.2
| M@
| 13/10
| +5
|-
| major 3-mosstep
| 13\29, 537.9
| M
| 15/11
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 16\29, 662.1
| N
| 19/13, 22/15
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 18\26, 744.8
| N&
| 20/13
| -5
|-
| dim. 5-mosstep
| 19\29, 786.2
| O@
| 11/7
| +6
|-
| perf. 5-mosstep
| 21\29, 869.0
| O
| 33/20, 38/23
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 24\29, 993.1
| P
| 16/9
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 26\28, 1075.9
| P&
| 13/7
| -4
|}


=== Hypohard ===
=== Hypohard tunings===
[[Hypohard]] tunings have [[step ratio]]s between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate [[8/7]], and the major 3-mosstep (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations. This set of JI approximations is associated with [[orgone]] temperament.
Hypohard smitonic tunings (2:1 to 3:1) have generators between 320{{c}} and 327{{c}}. The major 1-mosstep, or large step, tends to approximate [[8/7]] (231{{c}}) and the major 3-mosstep tends to approximate [[11/8]] (551{{c}}). [[26edo]] approximates these two intervals very well. These JI approximations are associated with [[orgone]] temperament.


Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
Other hypohard edos include [[11edo]] (not shown), [[15edo]] and [[37edo]].
The sizes of the generator, large step and small step of smitonic are as follows in various hypohard smitonic tunings.
{| class="wikitable right-2 right-3 right-4"
|-
!
! [[11edo]] (basic)
! [[15edo]] (hard)
! [[26edo]] (semihard)
! Some JI approximations
|-
| generator (g)
| 3\11, 327.27
| 4\15, 320.00
| 7\26, 323.08
| 77/64, 6/5
|-
| L (octave - 3g)
| 2\11, 218.18
| 3\15, 240.00
| 5\26, 230.77
| 8/7
|-
| s (4g - octave)
| 1\11, 109.09
| 1\15, 80.00
| 2\26, 92.31
| 128/121, (16/15)
|}
==== Intervals ====
Sortable table of major and minor intervals in hypohard smitonic tunings:
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable"|Degree
! [[11edo]] (basic)
! [[15edo]] (hard)
! [[26edo]] (semihard)
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-bgcolor="#eaeaff"
| unison
| 0\11, 0.0
| 0\15, 0.0
| 0\26, 0.0
| J
| 1/1
| 0
|-
| minor 1-mosstep
| 1\11, 109.1
| 1\15, 80.0
| 2\26, 92.3
| K@
|
| +4
|-
| major 1-mosstep
| 2\11, 218.2
| 3\15, 240.0
| 5\26, 230.8
| K
| 8/7
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 3\11, 327.3
| 4\15, 320.0
| 7\26, 323.1
| L
| 77/64, 6/5
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 4\11, 436.4
| 6\15, 480.0
| 10\26, 461.5
| L&
|
| -6
|-
| minor 3-mosstep
| 4\11, 436.4
| 5\15, 400.0
| 9\26, 415.4
| M@
| 14/11
| +5
|-
| major 3-mosstep
| 5\11, 545.5
| 7\15, 560.0
| 12\26, 553.9
| M
| 11/8
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 6\11, 656.6
| 8\15, 640.0
| 14\26, 646.2
| N
| 16/11
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 7\11, 763.6
| 10\15, 800.0
| 17\26, 784.62
| N&
| 11/7
| -5
|-
| dim. 5-mosstep
| 7\11, 763.6
| 9\15, 720.0
| 16\26, 738.5
| O@
|
| +6
|-
| perf. 5-mosstep
| 8\11, 872.7
| 11\15, 880.0
| 19\26, 876.9
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 9\11, 981.8
| 12\15, 960.0
| 21\26, 969.2
| P
| 7/4
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 10\11, 1090.9
| 14\15, 1120.0
| 24\26, 1107.7
| P&
|
| -4
|}


=== Parahard ===
{{MOS tunings|Step Ratios=3/1; 5/2; 7/3}}
In [[parahard]] smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach a perfect fifth. The parahard range contains very accurate edos such as [[53edo]] and [[72edo]], and has very accurate approximations to many [[low-overtone JI]] intervals, namely basic [[5-limit]] ratios and some ratios involving 13. However, the 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the [[4L 7s]] 11-note MOS, is suggested for getting 5-limit harmony.


This set of JI approximations is associated with [[kleismic]] temperament (we're specifically describing the 2.3.5.13 extension of it called [[Chromatic pairs#Cata|cata]]).
=== Parahard tunings ===
Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9{{c}} and 320{{c}}, putting it close to a pure 6/5 (316{{c}}). Stacking six generators and octave-reducing approximates 3/2 (702{{c}}), a diatonic perfect 5th, represented by the diminished 5-mosstep.


EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]].
This range contains very accurate edos such as [[53edo]] and [[72edo]], and has very accurate approximations to many [[low-overtone JI]] intervals, namely basic [[5-limit]] ratios and some ratios involving 13. However, 4L 3s only has one interval of 3/2, so it's suggested to use a larger MOS, such as [[4L 7s]], to achieve 5-limit harmony.


The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings (not including 15edo).
These JI approximations are associated with [[kleismic]] temperament, through the 2.3.5.13 extension known as [[Kleismic family#Cata|cata]].
{| class="wikitable right-2 right-3 right-4"
|-
!
! [[19edo]] (superhard)
! [[34edo]]  
! [[53edo]]
! JI intervals represented
|-
| generator (g)
| 5\19, 315.79
| 9\34, 317.65
| 14\53, 316.98
| 6/5
|-
| L (octave - 3g)
| 4\19, 252.63
| 7\34, 247.06
| 11\53, 249.06
| 15/13
|-
| s (4g - octave)
| 1\19, 63.16
| 2\34, 70.59
| 3\53, 67.92
| 25/24, 26/25
|}


==== Intervals ====
Parahard edos smaller than 53edo include [[15edo]] (not shown), [[19edo]], and [[34edo]].
Sortable table of major and minor intervals in parahard smitonic tunings:
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable"|Degree
! [[19edo]] (superhard)
! [[34edo]]
! [[53edo]]
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-bgcolor="#eaeaff"
| unison
| 0\19, 0.0
| 0\34, 0.0
| 0\53, 0.0
| J
| 1/1
| 0
|-
| minor 1-mosstep
| 1\19, 63.2
| 2\34, 70.6
| 3\53, 67.9
| K@
| 25/24, 26/25
| +4
|-
| major 1-mosstep
| 4\19, 252.6
| 7\34, 247.1
| 11\53, 249.1
| K
| 15/13
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 5\19, 315.8
| 9\34, 317.6
| 14\53, 317.0
| L
| 6/5
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 8\19, 505.3
| 14\34, 494.1
| 22\53, 498.1
| L&
| 4/3
| -6
|-
| minor 3-mosstep
| 6\19, 378.9
| 11\34, 388.2
| 17\53, 384.9
| M@
| 5/4
| +5
|-
| major 3-mosstep
| 9\19, 568.4
| 16\34, 564.7
| 25\53, 566.0
| M
| 18/13
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 10\19, 631.6
| 18\34, 635.3
| 28\53, 634.0
| N
| 13/9
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 16\19, 821.1
| 23\34, 811.8
| 39\53, 815.0
| N&
| 8/5
| -5
|-
| dim. 5-mosstep
| 11\19, 694.7
| 20\34, 705.9
| 31\53, 701.9
| O@
| 3/2
| +6
|-
| perf. 5-mosstep
| 14\19, 884.2
| 25\34, 882.4
| 39\53, 883.0
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 15\19, 947.4
| 27\34, 952.9
| 42\53, 950.9
| P
| 26/15
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 18\19, 1136.8
| 32\34, 1129.4
| 50\53, 1132.1
| P&
| 25/13
| -4
|}


== Modes ==
{{MOS tunings|Step Ratios=4/1; 11/3; 7/2}}
A naming scheme proposed by Alexandru Ianu ([[User:Ayceman]])<ref>Description of ''Sylvian Moon Dance'' mentioning the naming proposal https://musescore.com/user/36772625/scores/6700443 – The theme relates to the mystical nature of the Tribunal and TES lore, which fits smitonic.</ref>, relating to the Almsivi in Morrowind (TES):
{| class="wikitable center-all"
|-
! Mode
! [[Modal UDP Notation|UDP]]
! Name
|-
| LLsLsLs
| <nowiki>6|0</nowiki>
| Nerevarine
|-
| LsLLsLs
| <nowiki>5|1</nowiki>
| Vivecan
|-
| LsLsLLs
| <nowiki>4|2</nowiki>
| Lorkhanic
|-
| LsLsLsL
| <nowiki>3|3</nowiki>
| Sothic
|-
| sLLsLsL
| <nowiki>2|4</nowiki>
| Kagrenacan
|-
| sLsLLsL
| <nowiki>1|5</nowiki>
| Almalexian
|-
| sLsLsLL
| <nowiki>0|6</nowiki>
| Dagothic
|}
 
== Approaches ==
* [[4L 3s/Inthar's approach]]
 
== Temperaments ==
{{main|4L 3s/Temperaments}}
4L 3s has several temperament interpretations (see main article for mappings and optimal generator tunings):
 
# With generator size between 5\18 (333.3c) and 11\39 (338.5c): [[Sixix]], corresponding to a L/s ratio between 3/2 and 6/5.
# With generator size between 4\15 (320.0c) and 3\11 (327.3c): [[Orgone]], corresponding to a L/s ratio between 3 and 2.
# With generator size between 5\19 (315.8c) and 4\15 (320.0c): [[Kleismic]], corresponding to a L/s ratio between 4 and 3.
 
There are also other temperaments in the 4L 3s range, particularly [[amity]] and [[myna]], but 7 notes in the generator chain are not enough to contain the concordant chords optimized by these temperaments; you would need to use a [[MODMOS]] or use a larger MOS gamut, if you restrict to a rank-2 approach.


== Scales ==
== Scales ==
Line 1,004: Line 105:
* [[Cata7]]
* [[Cata7]]
* [[Myna7]]
* [[Myna7]]
== Scale tree==
{{MOS tuning spectrum
| 6/5 = [[Amity]]/[[hitchcock]]&nbsp;↑
| 5/4 = [[Sixix]]
| 4/3 = [[Supramin]]
| 13/8 = Golden 4L&nbsp;3s (868.3282{{c}})
| 12/5 = [[Hyperkleismic]]
| 5/2 = [[Orgone]]
| 13/5 = Golden superkleismic
| 8/3 = [[Superkleismic]]
| 11/3 = [[Hanson]]/[[keemun]]
| 6/1 = [[Oolong]]/[[myna]]&nbsp;↓
}}


== Music ==
== Music ==
Line 1,009: Line 124:
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [[File:Sixix Fugue.mp3]] A fugue in [[18edo]] smitonic functional harmony (WIP)


== Scale tree ==
== References ==
Generator ranges:
<references />
* Chroma-positive generator: 857.1429 cents (5\7) to 900 cents (3\4)
* Chroma-negative generator: 300 cents (1\4) to 342.8571 cents (2\7)


{| class="wikitable center-all"
[[Category:Smitonic|*]] <!--Main article-->
! colspan="6" rowspan="2" | Generator <br><small>(Chroma-positive)</small>
! colspan="2" | Cents
! rowspan="2" | L
! rowspan="2" | s
! rowspan="2" | L/s
! rowspan="2" | Comments
|-
! <small>Chroma-positive</small>
! <small>Chroma-negative</small>
|-
| 5\7 || || || || || || 857.143 || 342.857 || 1 || 1 || 1.000 ||
|-
| || || || || || 28\39 || 861.538 || 338.462 || 6 || 5 || 1.200 || [[Amity]]/[[hitchcock]]↑
|-
| || || || || 23\32 || || 862.500 || 337.500 || 5 || 4 || 1.250 || [[Sixix]]
|-
| || || || || || 41\57 || 863.158 || 336.842 || 9 || 7 || 1.286 ||
|-
| || || || 18\25 || || || 864.000 || 336.000 || 4 || 3 || 1.333 ||
|-
| || || || || || 49\68 || 864.706 || 335.294 || 11 || 8 || 1.375 ||
|-
| || || || || 31\43 || || 865.116 || 334.884 || 7 || 5 || 1.400 ||
|-
| || || || || || 17\58 || 865.574 || 334.426 || 10 || 7 || 1.429 ||
|-
| || || 13\18 || || || || 866.667 || 333.333 || 3 || 2 || 1.500 ||
|-
| || || || || || 47\65 || 867.692 || 332.308 || 11 || 7 || 1.571 ||
|-
| || || || || 34\47 || || 868.085 || 331.915 || 8 || 5 || 1.600 ||
|-
| || || || || || 55\76 || 868.421 || 331.579 || 13 || 8 || 1.625 || Golden smitonic (868.3282¢)
|-
| || || || 21\29 || || || 868.966 || 331.034 || 5 || 3 || 1.667 ||
|-
| || || || || || 50\69 || 869.565 || 330.435 || 12 || 7 || 1.714 ||
|-
| || || || || 29\40 || || 870.000 || 330.000 || 7 || 4 || 1.750 ||
|-
| || || || || || 37\51 || 870.588 || 329.422 || 9 || 5 || 1.800 ||
|-
| || 8\11 || || || || || 872.727 || 327.273 || 2 || 1 || 2.000 || Basic smitonic <br>(Generators smaller than this are proper)
|-
| || || || || || 35\48 || 875.000 || 325.000 || 9 || 4 || 2.250 ||
|-
| || || || || 27\37 || || 875.676 || 324.324 || 7 || 3 || 2.333 ||
|-
| || || || || || 46\63 || 876.190 || 323.810 || 12 || 5 || 2.400 || [[Hyperkleismic]]
|-
| || || || 19\26 || || || 876.923 || 323.077 || 5 || 2 || 2.500 || [[Orgone]] is in this region
|-
| || || || || || 49\67 || 877.612 || 322.388 || 13 || 5 || 2.600 || Golden superkleismic (877.7318¢)
|-
| || || || || 30\41 || || 878.049 || 321.951 || 8 || 3 || 2.667 || [[Superkleismic]]
|-
| || || || || || 41\56 || 878.571 || 321.429 || 11 || 4 || 2.750 ||
|-
| || || 11\15 || || || || 880.000 || 320.000 || 3 || 1 || 3.000 ||
|-
| || || || || || 36\49 || 881.633 || 318.367 || 10 || 3 || 3.333 ||
|-
| || || || || 25\34 || || 882.353 || 317.647 || 7 || 2 || 3.500 ||
|-
| || || || || || 39\53 || 883.019 || 316.981 || 11 || 3 || 3.667 || [[Hanson]]/[[keemun]] is in this region
|-
| || || || 14\19 || || || 884.211 || 315.789 || 4 || 1 || 4.000 ||
|-
| || || || || || 31\42 || 885.714 || 314.286 || 9 || 2 || 4.500 ||
|-
| || || || || 17\23 || || 886.957 || 313.043 || 5 || 1 || 5.000 ||
|-
| || || || || || 20\27 || 888.889 || 311.111 || 6 || 1 || 6.000 || [[Oolong]], [[myna]]↓
|-
| 3\4 || || || || || || 900.000 || 300.000 || 1 || 0 || → inf ||
|}
 
== References ==
[[Category:Smitonic|*]]<!--Main article-->
[[Category:7-tone scales]]
[[Category:7-tone scales]]
[[Category:Abstract MOS patterns]]
Retrieved from "https://en.xen.wiki/w/4L_3s"